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%% cosmology
%% ions / absorbers
\newcommand{\HI}{\hbox{{\sc H}{\sc i}} }
\newcommand{\NHI}{{N_{\rm HI}}}
\newcommand{\fNHI}{f(N_{\rm HI})}

\newcommand{\DXobs}{\Delta X^{\rm obs}}
\newcommand{\DXsim}{\Delta X^{\rm sim}}

\newcommand{\dXobs}{dX^{\rm obs}}
\newcommand{\dXsim}{dX^{\rm sim}}

\newcommand{\Ob}{\Omega_{\rm b} }
\newcommand{\Om}{\Omega_{\rm m} }
\newcommand{\OHI}{\Omega_{\rm HI} }
\newcommand{\OHH}{\Omega_{\rm H2} }
\newcommand{\Ol}{\Omega_{\Lambda} }
\newcommand{\Mpch}{h^{-1} \rm{\,Mpc} }
\newcommand{\kpch}{h^{-1} \rm{\,kpc} }
\newcommand{\Msunh}{h^{-1} \rm{\,M_{\odot}}}

% codes / simulations
\newcommand{\OWLS}{{\small OWLS} }

% journals

\newcommand{\mnras}{MNRAS}
\newcommand{\apjl}{ApJL}
\newcommand{\apj}{ApJ}
\newcommand{\apjs}{ApJS}
\newcommand{\aap}{A\&A}
\newcommand{\araa}{ARA\&A}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\title{Cosmological Absorption Systems}
\author{Gabriel Altay}
\begin{document}
\maketitle



\section{Absorption Distance}

A survey for absorption systems along lines of sight towards bright objects is characterized by the total redshift path searched for systems.   This redshift path is independent of a particular cosmological model.  If one chooses parameters that define a cosmology, one can transform from redshift path to absorption distance $X = \int dX$.  Absorption distance is defined such that random lines of sight will intersect a constant number of objects per absorption distance from  a population of aborbers with fixed proper size, and constant comoving number density \cite{1969ApJ...156L...7B}. 

\begin{eqnarray}
dX \equiv 
\frac{H_0}{c}(1+z)^3 c dt = 
\frac{H_0}{c}(1+z)^2 dx = 
\frac{H_0}{H(z)}(1+z)^2 dz =
\frac{(1+z)^2}{E(z)} dz 
\end{eqnarray}
where $dx$ is the comoving distance, $H_0$ is the Hubble parameter today, and $H(z)$ is the Hubble parameter as a function of redshift, 

\begin{eqnarray}
H_0 &=& 100  \; h {\rm \frac{km}{s \,Mpc} } 
= 3.2407789 \times 10^{-18} \; h {\rm \,s^{-1} } 
\end{eqnarray}
In a flat universe with a cosmological constant, the redshift dependence of the Hubble parameter can be written,

\begin{eqnarray}
H(z) &=& H_0 E(z) = H_0 \sqrt{\Om(1+z)^3 + \Ol}
\end{eqnarray}
which can be analytically integrated to find the absorption distance over a redshift interval.

\begin{eqnarray}
\int_{z_1}^{z_2} dX = 
\int_{z_1}^{z_2} \frac{(1+z)^2 dz }{\sqrt{\Om(1+z)^3 + \Ol}} =
\frac{2}{3} \frac{\sqrt{\Om(1+z)^3 + \Ol}}{\Om} \bigg|_{z_1}^{z_2} = 
\frac{2}{3} \frac{E(z)}{\Om} \bigg|_{z_1}^{z_2}
\end{eqnarray}

For simulation boxes that are small enough for redshift evolution to be ignored across the box, a simple calculation gives the absorption distance for a single line of sight parallel to the box axes in terms of the box length, $\Delta L$, in $\Mpch$.

\begin{eqnarray}
\Delta X_1 &=& \frac{H_0}{c}(1+z)^2 \Delta L = 
\frac{100 \,h {\rm  \,s^{-1}  \,km  \,Mpc^{-1}}}
{2.9979 \times 10^5 {\rm \,km \,s^{-1}}}
(1+z)^2 \Delta L \nonumber \\
&=& 3.3357 \times 10^{-4} \,h {\rm \,Mpc^{-1}} (1+z)^2 \Delta L
\end{eqnarray}
For example, a box with $\Delta L = 25 \, \Mpch$ (like many of the {\small OWLS} boxes) at $z=3$ has an absorption distance of $\Delta X = 0.1334$. 


\section{Comparing Observations and Simulations}

\subsection{Column Density Distribution Function}

One of the fundamental observables in absorption line studies is the $\HI$ column density distribution function, $\fNHI$. This quantity is most often defined by observers as, 
\begin{eqnarray}
  \fNHI = \frac{m}{\Delta \NHI \times \sum_i^n \Delta X_i} = 
  \frac{m}{\Delta \NHI \Delta X}
\end{eqnarray}
where $m$ is the number of absorbers with column densities between $\NHI$ and $\NHI + \Delta \NHI$, and $\sum_i^n \Delta X_i$ is a sum over the absorption distances of all lines of sight surveyed for absorbers.  The factor of $\Delta X$ in the denominator is dependent on cosmological parameters, and so if the observations assumed a different set than the simulation, we have to correct for that. 
What we have is,
\begin{eqnarray}
\frac{\dXsim}{\dXobs} = 
\frac{E^{\rm obs}(z)}{E^{\rm sim}(z)} = 
\sqrt{ \frac{\Om^{\rm obs}(1+z)^3 + \Ol^{\rm obs}}
            {\Om^{\rm sim}(1+z)^3 + \Ol^{\rm sim}} } \approx
\sqrt{ \frac{\Om^{\rm obs} }
            {\Om^{\rm sim} } }
\end{eqnarray}
where we have used the high $z$ approximation of $E(z)$ to eliminate the $z$ dependence.  With this approximation, 
\begin{eqnarray}
\frac{\DXobs}{\DXsim} = \sqrt{ \frac{\Om^{\rm sim} }{\Om^{\rm obs} } } 
\end{eqnarray}
This means if we know the value of $\fNHI$ assuming one cosmology chosen by the observer, and we want to convert it to our simulation cosmology for comparison we use the following relationship, 
\begin{eqnarray}
f^{\rm sim} = \frac{m}{\Delta \NHI \Delta X^{\rm sim} }
= \frac{m}{\Delta \NHI \Delta X^{\rm obs} }
\frac{\Delta X^{\rm obs}}{\Delta X^{\rm sim}} = 
f^{\rm obs} \sqrt{ \frac{\Om^{\rm sim} }{\Om^{\rm obs} } }
\end{eqnarray}
Alternatively, if you have access to the actual redshift intervals that make up the sample, you can redo the integration and leave the $z$-dependent pieces from Eq. 6 in. 


\subsection{Zeroth Moment}

The zeroth moment of $\fNHI$ is known as the incidence, $\ell(X)$.  Let $w = \log \NHI$ then $d\NHI = \ln(10) \NHI dw$.
\begin{eqnarray}
 \ell(X) &=& \int_{N_1}^{N_2} \fNHI d\NHI \\
 &=& \int_{w_1}^{w_2} \ln(10) f(w) \NHI dw \\
 &=& \int_{w_1}^{w_2} \ln(10) f(w) 10^w dw \\
 &\approx& \frac{1}{\Delta X} \sum_i^n m_i
\end{eqnarray}
where the summation is over the $\NHI$ bins.

\subsection{First Moment}

\subsubsection{Theory}

The first moment of $\fNHI$ is $\OHI$ defined as, 
\begin{eqnarray}
\OHI &=& \frac{\mu m_{\rm H} H_0}{c \rho_{c,0}}
\int_{N_1}^{N_2} \NHI \fNHI d\NHI, \quad
\rho_{c,0} = \frac{3 H^2_0}{8 \pi G} \\
&=& \frac{8 \pi G m_{\rm H}}{3 c} \frac{\mu}{H_0}
\int_{N_1}^{N_2} \NHI \fNHI d\NHI \\
&\approx& \frac{8 \pi G m_{\rm H}}{3 c}
\frac{\mu}{H_0 \Delta X}
\sum_i^n N^i_{\rm HI} m_i
\end{eqnarray}
In the majority of papers I've read, observers adopt a value for the mean molecular mass of $\mu = 1.3$.  This value is approximately what you would get for a neutral mixture of Hydrogen and Helium with the Hydrogen mass fraction set at the cosmological value of about 0.75. If you are interested in just neutral Hydrogen (as we are) then you should use a value of $\mu = 1.0$ as for all the systems that contribute appreciably to $\OHI$ the Hydrogen is predominantly neutral.  We also must take account of different values for $H_0$ and $\Delta X$. 

\begin{eqnarray}
  \Omega^{\rm sim}_{\rm HI} &=& \Omega^{\rm obs}_{\rm HI}
  \frac{\mu^{\rm sim}}{\mu^{\rm obs}}
  \frac{H_0^{\rm obs}}{H_0^{\rm sim}}
  \frac{\Delta X^{\rm obs} }{\Delta X^{\rm sim} }  \\
  &=& \Omega^{\rm obs}_{\rm HI}
  \frac{\mu^{\rm sim}}{\mu^{\rm obs}}
  \frac{H_0^{\rm obs}}{H_0^{\rm sim}}
  \sqrt{ \frac{\Om^{\rm sim} }{\Om^{\rm obs} } } \\
  &=& \Omega^{\rm obs}_{\rm HI} f_{\mu} f_H f_{\Omega}
\end{eqnarray}




\subsubsection{Numerical Work}


Here we simply tabulate the spatial and mass resolutions used in various simulations. 

%%
%% table of resolutions used in various simulations
%%


\vspace{7 mm}
\begin{tabular}{| c | c | c | c | c | c | c |}
  \hline                       
  Reference & Model Name & L          & N & 
  $m_{\rm b}$ & $m_{\rm d}$ & $\epsilon_{\rm com}$  
  \\
            &            & [$\Mpch$]  &   &  
  $\Msunh$  & $\Msunh$   & $\kpch$      \\

  \hline
  \hline

  \cite{1992ApJ...399L.109K} &
  NA & 
  11 & $32^3$ & $5.8 \times 10^8$ & $1.1 \times 10^{10}$ & 10.0 \\

  \hline
  \hline

  \cite{1996ApJ...457L..57K} &
  NA & 
  11 & $64^3$ & $7.5 \times 10^7$ & $1.4 \times 10^{9}$ & 10.0 \\

  \hline
  \hline

  \cite{2010MNRAS.402.1536S} &
  {\it REF\_L025N512} & 
  25 & $512^3$ & $1.4 \times 10^6$ & $6.3 \times 10^6$ & 1.95 \\

  \hline
  \hline

  \cite{2010arXiv1008.4242H} &
  nw &
  8 & $256^3$ & $4.84 \times 10^5$ & $3.15 \times 10^6$ & 0.625  
\\
  \cite{2010arXiv1008.4242H} &
  cw &
  8 & $256^3$ & $4.84 \times 10^5$ & $3.15 \times 10^6$ & 0.625  
\\
  \cite{2010arXiv1008.4242H} &
  vzw &
  8 & $256^3$ & $4.84 \times 10^5$ & $3.15 \times 10^6$ & 0.625  
\\

  \hline
  \hline


\end{tabular}



Here we simply tabulate the cosmological parameters used in various simulations. 

%%
%% table of cosmologies used in various simulations
%%

\vspace{7 mm}
\begin{tabular}{| c | c | c | c | c | c | c |}
  \hline                       
  Reference & Model Name & $100 \{ \Om,\Ol,\Ob \} $ &
  $H_0$ & $\sigma_{\rm 8}$  
  \\
  &   &   & [km/s/Mpc]  & [$\Mpch$]  
  \\

  \hline
  \hline

  \cite{1992ApJ...399L.109K} &
  NA & 
  \{100,0,5.0\} & 50.0 & 0.7 \\

  \hline
  \hline

  \cite{1996ApJ...457L..57K} &
  NA & 
  \{100,0,5.0\} & 50.0 & 0.7 \\

  \hline
  \hline

  \cite{2010MNRAS.402.1536S} &
  {\it REF\_L025N512} & 
  $\{ 23.8, 76.2, 4.18 \}$ & 73.0 & 0.74 
  \\

  \hline
  \hline

  \cite{2010arXiv1008.4242H} &
  nw &
  $\{ 30.0, 70.0, 4.00 \}$ & 70.0 & 0.90 
\\
  \cite{2010arXiv1008.4242H} &
  cw &
  $\{ 30.0, 70.0, 4.00 \}$ & 70.0 & 0.90 
\\
  \cite{2010arXiv1008.4242H} &
  vzw &
  $\{ 30.0, 70.0, 4.00 \}$ & 70.0 & 0.90 
\\

  \hline



\end{tabular}





Here we simply tabulate the wind parameters used in various simulations. 

%%
%% table of wind parameters
%%


\vspace{7 mm}
\begin{tabular}{| c | c | c | c |}
  \hline                       
  Reference & Model Name & $\eta$ & $v_{\rm w}$ 
  \\
            &            &   & [km/s]   
\\

  \hline
  \hline

  \cite{1992ApJ...399L.109K} &
  NA & NA & NA  \\

  \hline
  \hline

  \cite{1996ApJ...457L..57K} &
  NA & NA & NA \\

  \hline
  \hline

  \cite{2010MNRAS.402.1536S} &
  {\it REF\_L025N512} & 
  2 & 600 \\

  \hline
  \hline

  \cite{2010arXiv1008.4242H} &
  nw &
  NA & NA 
\\
  \cite{2010arXiv1008.4242H} &
  cw &
  2 & 484 
\\
  \cite{2010arXiv1008.4242H} &
  vzw &
  $\propto \sigma^{-1}$ & $\propto \sigma$   
\\

  \hline
  \hline


\end{tabular}





\subsubsection{Observations}

The \OWLS cosmology is $\{\Ol=0.762,\, \Om=0.238,\, H_0=73\}$.  Therefore $h_{75} = 73/75 = 0.973$ and $h^{-1}_{75} = 75/73 = 1.03$.  For observations involving the observers favorite $\{\Ol=0.7,\, \Om=0.3,\, H_0=70\}$, ``737'' cosmology that report $\Omega_{\rm g}$ instead of $\OHI$ we have  
$f_{\mu} = 1/1.3 = 0.769$, 
$f_H = 70/73 = 0.959$, 
$f_{\Omega} = \sqrt{0.238/0.300} = 0.891$.  
Putting all the factors together we have $f_{\mu} f_{H} f_{\Omega} = 0.657$. 
For \cite{2009A&A...508..133G} we have $f_{H} = 72/73 = 0.986$ and
$f_{\mu} f_{H} f_{\Omega} = 0.676$.

\vspace{7 mm}
\begin{tabular}{| c | c | c | c | c | c |}
  \hline                       
  Source & Reported Value & $<z>$ & Cosmology         & f & \OWLS Value\\
         &                &       & $\{\Ol,\Om,H_0\}$ &   & $\times 10^{-4}$ \\

  \hline
  \hline

  \cite{2005MNRAS.359L..30Z} & 
  $\OHI = (3.5 \pm 0.4 \pm 0.4) \times 10^{-4} h^{-1}_{75}$ & 
  0 &
  - & 
  1.03 & 
  $(3.6 \pm 0.82)$ \\

  \hline

  \cite{2006ApJ...636..610R} &
  $\Omega_{\rm g}^{\rm DLA} = (9.7 \pm 3.6) \times 10^{-4}$ &
  0.609 &
  $\{0.7,0.3,70\}$ &
  0.657 & 
  $(6.4 \pm 2.4) $\\

  \cite{2006ApJ...636..610R} &
  $\Omega_{\rm g}^{\rm DLA} = (9.4 \pm 2.8) \times 10^{-4}$ &
  1.219 &
  $\{0.7,0.3,70\}$ &
  0.657 & 
  $(6.2 \pm 1.8) $\\

  \hline

  \cite{2009A&A...505.1087N} & 
  $\Omega_{\rm g}^{\rm DLA} = (0.82 \pm 0.09) \times 10^{-3}$  & 
  2.44 &
  $\{0.7,0.3,70\}$ & 
  0.657 & 
  $(5.4 \pm 0.59) $\\

  \cite{2009A&A...505.1087N} & 
  $\Omega_{\rm g}^{\rm DLA} = (0.85 \pm 0.09) \times 10^{-3}$  & 
  2.74 &
  $\{0.7,0.3,70\}$ & 
  0.657 & 
  $(5.6 \pm 0.59) $\\

  \cite{2009A&A...505.1087N} & 
  $\Omega_{\rm g}^{\rm DLA} = (1.03 \pm 0.10) \times 10^{-3}$  & 
  3.02 &
  $\{0.7,0.3,70\}$ & 
  0.657 & 
  $(6.8 \pm 0.66) $\\

  \cite{2009A&A...505.1087N} & 
  $\Omega_{\rm g}^{\rm DLA} = (1.29 \pm 0.15) \times 10^{-3}$  & 
  3.49 &
  $\{0.7,0.3,70\}$ & 
  0.657 & 
  $(8.5 \pm 0.99) $\\

  \hline

  \cite{2009A&A...508..133G} &
  $\Omega_{\rm g}^{\rm DLA} = (1.43 \pm 0.33) \times 10^{-3}$ &
  3.168 &
  $\{0.7,0.3,72\}$ &
  0.676 & 
  $(9.7 \pm 2.2) $\\

  \cite{2009A&A...508..133G} &
  $\Omega_{\rm g}^{\rm DLA} = (1.41 \pm 0.26) \times 10^{-3}$ &
  3.618 &
  $\{0.7,0.3,72\}$ &
  0.676 & 
  $(9.5 \pm 1.8) $\\

  \cite{2009A&A...508..133G} &
  $\Omega_{\rm g}^{\rm DLA} = (0.97 \pm 0.22) \times 10^{-3}$ &
  4.048 &
  $\{0.7,0.3,72\}$ &
  0.676 & 
  $(6.6 \pm 1.5) $\\

  \hline

  \cite{2009A&A...508..133G} &
  $\Omega_{\rm g}^{\rm DLA+sub} = (1.71 \pm 0.33) \times 10^{-3}$ &
  3.168 &
  $\{0.7,0.3,72\}$ &
  0.676 & 
  $(11.6 \pm 2.2) $\\

  \cite{2009A&A...508..133G} &
  $\Omega_{\rm g}^{\rm DLA+sub} = (1.65 \pm 0.26) \times 10^{-3}$ &
  3.618 &
  $\{0.7,0.3,72\}$ &
  0.676 & 
  $(11.2 \pm 1.8) $\\

  \cite{2009A&A...508..133G} &
  $\Omega_{\rm g}^{\rm DLA+sub} = (1.21 \pm 0.22) \times 10^{-3}$ &
  4.048 &
  $\{0.7,0.3,72\}$ &
  0.676 & 
  $(8.2 \pm 1.5) $\\


  \hline  

\end{tabular}



\subsection{Contribution to $\OHI$ from different log $\NHI$ bins}

\begin{eqnarray}
\frac{d \OHI}{d \log \NHI} = 
\ln(10) \times 10^{2 \log \NHI} \fNHI
\end{eqnarray}




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\end{document}
